Problem 23 - Algebra 1

Question

Write an equation in point-slope form of the line that passes through the given points. $$ (-9,10),(-4,-3) $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the points (-9,10) and (-4,-3) is \(y - 10 = \frac{-13}{5}(x + 9)\).

Step 1: Identify the given points

The given points from the problem are (-9,10) and (-4,-3).

Step 2: Calculate the slope

The formula for the slope (m) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). If we consider (-9,10) as Point1 (\(x_1,y_1\)) and (-4,-3) as Point2 (\(x_2,y_2\)), we substitute into the formula to get \(m = \frac{-3 - 10}{-4 - (-9)} = \frac{-13}{5}\).

Step 3: Write the equation in point-slope form

In point-slope form the equation of a line is \(y - y_1 = m(x - x_1)\). We substitute m, \(x_1\) and \(y_1\), which are -13/5, -9 and 10 respectively, to get the equation \(y - 10 = \frac{-13}{5}(x + 9)\).

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is essential in algebra and coordinate geometry, as it tells us about the 'steepness' and the direction of the line. The slope is represented by the letter 'm' and calculated using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

To find the slope between two points, simply subtract the y-coordinate of the first point from the y-coordinate of the second point and divide it by the subtraction of the x-coordinate of the first point from the x-coordinate of the second point. This value can be positive or negative, which indicates the direction of the line. A positive slope means the line is ascending from left to right, while a negative slope indicates a descending line.

For example, given two points \[(-9,10), (-4,-3)\], we calculate the slope by taking the y-coordinates and x-coordinates of these points and applying them to our formula: \(m = \frac{-3 - 10}{-4 - (-9)} = \frac{-13}{5}\). The resulting slope, \(\frac{-13}{5}\), tells us the line falls 13 units vertically for every 5 units it moves horizontally.

Linear Equations

Linear equations form the basis of much of algebra and can be visualized as straight lines when plotted on a graph. They take the general form \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept, the point where the line crosses the y-axis.

The point-slope form of a linear equation is a variation that is particularly useful when you have a known point on the line and its slope. It is expressed as \(y - y_1 = m(x - x_1)\), where \(x_1\) and \(y_1\) are the coordinates of the known point, and 'm' is the slope of the line. This form makes it straightforward to write the equation if you're given a point and the slope. Once the equation is established, you can rearrange it to the slope-intercept form or standard form, depending on what's more suitable for the problem at hand.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves graphing and understanding the properties of shapes using a coordinate system. In this system, every point on the plane is defined by an x-coordinate and a y-coordinate. The relationship between the coordinates is described using algebraic equations.

A fundamental concept in coordinate geometry is that of the Cartesian coordinate system, consisting of a horizontal axis (x-axis) and a vertical axis (y-axis), which intersect at the origin (0,0). Lines can be plotted on this system, with each point on the line representing a solution to its equation.

Points, lines, and other shapes graphed on the coordinate plane can be analyzed to find distances, midpoints, slopes, and other geometric properties. Using the principles of coordinate geometry, we can solve real-world problems, like finding the shortest distance between two points or determining the intersection of two lines.